* " Proof " ( I Am Kloot song ), a song by I Am Kloot from their eponymous second album

When a second child is cut, the node itself needs to be cut from its parent and becomes the root of a new tree ( see Proof of degree bounds, below ).

This book inspired Douglas Hofstadter to take up the study of mathematical logic, write his famous book Gödel, Escher, Bach, and prepare a second edition of Gödel's Proof, published in 2002.

Echo had originally planned to release " Proof " as a single to promote the second album, but after agreeing on artwork and a track listing ( and commissioning a video featuring Christopher Eccleston, directed by Krishna Stott ), the label shelved these ideas and subsequently issued the single as a download only ( though a few copies were issued in Europe on the PIAS imprint ).

The second single from Proof of Youth " Doing It Right " was released on 3 September 2007 and received day time radio play from BBC Radio 1 and soundtracked the BBC's launch of the iPlayer.

In May 2011, it was announced that Body of Proof was renewed for a second season, which start Fall 2011.

Proof of first property: from 3. we obtain ( a < sup >− 1 </ sup >)< sup >− 1 </ sup >=

Proof: Notice that the nim-sum (⊕) obeys the usual associative and commutative laws of addition (+), and also satisfies an additional property, x ⊕ x = 0 ( technically speaking, the nonnegative integers under ⊕ form an Abelian group of exponent 2 ).

Proof of such a property interest would not, of course, entitle him to reinstatement.

Proof of this is the legacy " coral " that led to the U. S., especially since the founding of St. Olaf College Choir by F. Melius Christiansen in 1912.

Proof: If is diagonalizable, then A is annihilated by some polynomial, which has no multiple root ( since ) and is divided by the minimal polynomial of A.

The dome is " Water Proof ", because Sandy does not breathe underwater since she is a squirrel.

Proof: Using the same sum for R ( p, n ) as in Lemma 2, we see that since p < sup > 2 </ sup > > 2n, in fact only the first term is nonzero ; this term is exactly the right-hand side of the above equation.

Also from the Gang Starr Foundation, Premier produced and supervised Group Home's Livin ' Proof ; although overlooked at the time of its 1995 release, the album has since come to find similar acclaim.

She was a regular on the science fiction series Dark Skies ( 1997 ) and the legal drama series Shark ( 2006 – 08 ), Since 2011, she has starred as Dr. Kate Murphy in the ABC drama series Body of Proof and since 2009 she has played occasional guest star Tara Cole on Leverage.

He rose to fame by producing Reservoir Dogs in 1992 and has since produced all of Quentin Tarantino's films with the exception of Death Proof and Django Unchained.

Positioned alongside the BSA factory sportsground and Sparkbrook, from which the local area gets its name, the Parker-Hale factory headquarter building echoes the traditions of the Birmingham gunmaking industry, since it occupies the very site that the Proof House located their testing range.

Music critic Mark Deming of Allmusic praised the album and was " easily T-Bone Burnett's strongest album since Proof Through the Night, and a rare pleasure for thinking music fans.

#* Proof: If q divides 2 < sup > p </ sup > − 1 then 2 < sup > p </ sup > ≡ 1 ( mod q ).

Proof: Let d be the position of the leftmost ( most significant ) nonzero bit in the binary representation of s, and choose k such that the dth bit of x < sub > k </ sub > is also nonzero.

< div class =" NavHead " style =" background :# ccf ; text-align: left ; font-size: larger ;"> Proof of ( 1 )</ div >

Proof: Since p is a prime number the only possible values of gcd ( p < sup > k </ sup >, m ) are 1, p, p < sup > 2 </ sup >, ..., p < sup > k </ sup >, and the only way for gcd ( p < sup > k </ sup >, m ) to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to p < sup > k </ sup > are p, 2p, 3p, ..., p < sup > k − 1 </ sup > p = p < sup > k </ sup >, and there are p < sup > k − 1 </ sup > of them.

( Proof: If m < n, then we can view R < sup > m </ sup > as a subspace of R < sup > n </ sup >, and the non-empty open subsets of R < sup > m </ sup > are not open when considered as subsets of R < sup > n </ sup >.

Proof: It is straightforward to prove that ∇ satisfies the Leibniz identity and is C < sup >∞</ sup >( S < sup > 2 </ sup >) linear in the first variable.

: Proof: If we assume, w. l. o. g., Q < sub > A </ sub >∩ Q < sub > B </ sub > is empty then L ( A )∪ L ( B ) is recognized by the Büchi automaton ( Q < sub > A </ sub >∪ Q < sub > B </ sub >, Σ, Δ < sub > A </ sub >∪ Δ < sub > B </ sub >, I < sub > A </ sub >∪ I < sub > B </ sub >, F < sub > A </ sub >∪ F < sub > B </ sub >).

#* Proof: Then so Thus However, is prime, so or In the former case, hence ( which is a contradiction, as neither 1 nor 0 is prime ) or In the latter case, or If however, which is not prime.

#* Proof: suppose that p is composite, hence can be written with a and Then is prime, but and contradicting statement 1.

( Proof: either 1 is positive or − 1 is positive.

Fourth Session ( October 1, 787 ) -- Proof of the lawfulness of the veneration of icons was drawn from Exodus 25: 19 sqq.

* Arora, Sanjeev ; Barak, Boaz, " Complexity Theory: A Modern Approach ", Cambridge University Press, 2009, ISBN 978-0-521-42426-4, section 1. 4, " Machines as strings and the universal Turing machine " and 1. 7, " Proof of theorem 1. 9 "

*: Puncture Proof: 1

Proof of Lemma 1:

* 3200th Proof Test Group, 1 July 1948-1 July 1953

* 3200th Proof Test Wing, 1 April 1951-1 July 1952

* Proof that the Sample Bivariate Correlation Coefficient has Limits ± 1

# is a polynomial ( of degree ≤ n − 1 ) in .< ref > Proof: When A is normal, use Lagrange's interpolation formula to construct a polynomial P such that, where are the eigenvalues of A .</ ref >

From the criterion it also follows that Carmichael numbers are cyclic .< ref > Proof sketch: If is square-free but not cyclic, for two prime factors and of.

::< ref > Proof: Apply the rank-nullity theorem to the inequality:

::< ref > Proof: The map

Proof: Suppose that and are two identity elements of.

Proof: Suppose that and are two inverses of an element of.

Among Moore's most famous works are his book Principia Ethica, and his essays, " The Refutation of Idealism ", " A Defence of Common Sense ", and " A Proof of the External World ".

He famously put the point into dramatic relief with his 1939 essay " Proof of an External World ", in which he gave a common sense argument against scepticism by raising his right hand and saying " Here is one hand ," and then raising his left and saying " And here is another ," then concluding that there are at least two external objects in the world, and therefore that he knows ( by this argument ) that an external world exists.

Proof of payment is often in the form of adhesive postage stamps, but postage meters are also used for bulk mailing.

Note that the relationships above are only valid for the form of the Fourier transform shown in the Proof section below.

Proof of credentials are verified before the event or on-site.

In the shortest of them ( 43 pages as of 2009 ), which he titles " Apology for the Proof of the Riemann Hypothesis " ( using the word " apology " in the rarely used sense of apologia ), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-functions ( thus proving GRH ) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple.

* A 2011 episode of the series Body of Proof features character Dana Delany investigating a homicide in a neighborhood in which all neighbors are in the Lifestyle.

Distributions of the Mizar Proof Checker for all major operating systems are freely available for downloaded at the Mizar Project website.

Proof of her feelings are further cemented when she moves to kiss Kaidou, asking why, if he wanted to marry into the profession, didn't he choose to marry her due to her father's political influence.

Proof methods that are not direct include proof by contradiction, including proof by infinite descent.

The edge lettering does not occur at the same time as the minting of the coins, allowing for the natural occurrence of the lettering in either orientation, except Proof Coins where the date and lettering are all " right-side-up.

Proof by Contradiction: Assume that there is a non-empty set of natural numbers that are not interesting.

Proof of this are the results of UDI in the 2000 municipal elections, the parliamentary elections of 2001, and the Lagos-Longueira agreement of January 17, 2003 to modernize the State administration and give a consensual political solution to Inverlink case and MOP-Gate case, which affected the institutional stability of the Lagos administration.

Proof mintages are indicated in italics.

Although not suggested by natural language, and in contrast to usage in other areas, Proof of Principle and Proof of Concept are not synonymous in drug development.